#8534876

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#41468	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4032 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,11)(4,13)(5,10)(6,7)(8,12)(9,14)(15,16)(33,34,37,40)(35,38,42,46)(36,43,44,39)(41,47,48,45);(2,7)(3,12)(9,13)(10,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,31)(28,29)(30,32)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,7)(4,14)(8,11)(10,16)(17,19,22,24)(18,21,25,26)(20,28,30,27)(23,31,32,29)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,3)(2,8)(4,10)(5,9)(6,12)(7,11)(13,15)(14,16)(17,18)(19,21)(20,23)(22,25)(24,26)(27,29)(28,31)(30,32)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,4,6,14)(2,9,7,13)(3,10,12,16)(5,11,15,8)(17,20,22,30)(18,23,25,32)(19,27,24,28)(21,29,26,31)(33,36,37,44)(34,39,40,43)(35,41,42,48)(38,45,46,47);(1,5)(2,10)(3,13)(4,11)(6,15)(7,16)(8,14)(9,12)(17,21)(18,24)(19,25)(20,29)(22,26)(23,28)(27,32)(30,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8442072);